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Looking for a proof in the literature of the following lemma:

Let $K\subset\mathbb{R}^d$ be a bounded domain. Let $P_X$ be a finite dimensional subspace of $\mathcal{H}^k(K)$ then

$$\left\|v\right\|_{\mathcal{H}^k(K)} \leq C diam(K)^{-k} \left\|v\right\|_{L_2(K)}$$

for all $v\in P_X$ and where the constant $C$ does not depend on $diam(K)$.

There is a proof in The mathematical theory of finite element method By Susanne C. Brenner, L. Ridgway Scott p111 but they do not check if the constant $C$ is independent of $diam(K)$.

Thanks in advance.

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Having trouble getting the main equation to Latex... – alext87 Feb 23 '11 at 14:56
Theorem II.6.8 in Braess' book is quite closely related to your question, but the proof is worked out only for very special domains coming from finite element methods. But maybe you can use the idea... – András Bátkai Feb 23 '11 at 22:26

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